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I was considering a situation where a bullet was shot at an angle and lodges itself into a block and then the to move as a system. As usual, I figured I could solve for the final velocity of the system using conservation of momentum and then to check my answer, I figured that I'd use conservation of energy to confirm the total velocity that I got using momentum principles. These were very different. I thought that maybe the issue would be with the conservation of energy here since there is no external force on the system the conservation of momentum should hold. Is there an assumption that I'm making that is wrong here? If so, please explain why it is wrong.

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When the bullet lodges, a large part of the original kinetic energy is converted into heat (and also sound etc), so you can't use conservation of energy to solve the problem. Such collisions are called inelastic collisions. Elastic collisions are when the kinetic energy is conserved. Intuitively, they are when the objects bounce perfectly off each other.

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